SOLVED! My Ring 3 was supposed to work with my existing digital doorbell model, but did not.
After trying everything under the sun (new transformer, etc.) it turned out all I had to do was REVERSE THE WIRES coming into my Ring from the doorbell. The Ring 3 has a built in diode which the digital doorbell needs in line at the doorbell connection, and it NEEDS TO BE in the right “polarity” so to speak. I tried this as a last resort (tech support made no mention of this, nor is it in the written instructions) and it worked perfectly! Hope this helps others who have had this issue, and I really think Ring needs to inform people of this issue right out of the gate in the installation instructions.